Factor the following expression: $-8$ $x^2+$ $31$ $x$ $-21$
Solution: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-8)}{(-21)} &=& 168 \\ {a} + {b} &=& & & {31} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $168$ and add them together. The factors that add up to ${31}$ will be your ${a}$ and ${b}$ When ${a}$ is ${7}$ and ${b}$ is ${24}$ $ \begin{eqnarray} {ab} &=& ({7})({24}) &=& 168 \\ {a} + {b} &=& {7} + {24} &=& 31 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {-8}x^2 +{7}x +{24}x {-21} $ Group the terms so that there is a common factor in each group: $ ({-8}x^2 +{7}x) + ({24}x {-21}) $ Factor out the common factors: $ x(-8x + 7) - 3(-8x + 7) $ Notice how $(-8x + 7)$ has become a common factor. Factor this out to find the answer. $(-8x + 7)(x - 3)$